Lcm of degrees of irreducible representations

This article defines an arithmetic function on groups
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This term is related to: linear representation theory
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Definition

For a group over a field

Suppose $G$ is a group and $K$ is a field. The lcm of degrees of irreducible representations of $G$ is defined as the least common multiple of all the degrees of irreducible representations of $G$ over $K$ .

Typical context: finite group and splitting field

The typical context is where $G$ is a finite group and $K$ is a splitting field for $G$ . In particular, the characteristic of $K$ is either zero or is a prime not dividing the order of $G$ , and every irreducible representation of $G$ over any extension field of $K$ can be realized over $K$ .

Note that the lcm of degrees of irreducible representations depends (if at all) only on the characteristic of the field $K$ . This is because the degrees of irreducible representations over a splitting field depend only on the characteristic of the field.

Default case: characteristic zero

By default, when referring to the lcm of degrees of irreducible representations, we refer to the case of characteristic zero, and we can in particular take $K=\mathbb {C}$ .

Related facts

What it divides

Any divisibility fact stating that the degree of every irreducible representation over a splitting field must divide some fixed number implies that the lcm also divides that fixed number. Some of these are listed below:

Degree of irreducible representation divides order of group: Hence, the lcm of degrees of irreducible representations divides the order of the whole group.
Degree of irreducible representation divides index of center: Hence, the lcm of degrees of irreducible representations divides the index of the center, which is also the order of the inner automorphism group.
Degree of irreducible representation divides index of abelian normal subgroup: Hence, the lcm of degrees of irreducible representations divides the index of any abelian normal subgroup.

Subgroups, quotients, direct products